Suff Automata

-

,

. .

: . .

-

2007

2

3

1. 4

1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2. . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3. . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4. . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5. . . . . . . . . . . . . . . . . . . . . . . . 9

1.6. . . . . . . . . . . . . . . . . . . . . . . 10

1.7. . . . . . . . . . . . . . . . . . 12

2. -

15

2.1. . . . . . . 15

2.2. 16

2.3. . . 17

2.4. 19

2.5. 22

3.

25

3.1. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1. . . . . . . . . . . . . . . . . . . . . 26

3.1.2. . . . . . . . . . . . . . . . . . . . . 27

3.1.3. . . . . . . . . . . . . . . . . . . . 27

3.1.4. . . . . . . . . . . . . . . . . 27

3.2. . . . 28

34

35

3

-

[1]. -

[1]

.

-

, -

[1]. ,

[2], - [3] [4]

[1] .

,

.

. ,

(-

[5],

- [5]),

-

.

, -

,

, .

,

-

. ,

.

4 1.

1.1.

1. [5, . 3] , -

.

. .

+ = \ {}.

2. [5, . 3] w, |w|

w, w.

3. [5, . 4] w

u, x y, u = xwy.

4. [5, . 4] w

u, y, u = wy.

5. [5, . 4] w

u, x, u = xw.

6. x

Fact(x).

7. x

Suff(x).

8. [5, . 4] k w

w[1 . . . k].

9. k- w x[k].

51.2.

10. [5, . 116] x

y Ry(x) = x1Suff(y) = {s | xs Suff(y)}.

, x y

y,

x.

11. [5, . 116] u v -

y (u y v), Ry(u) = Ry(v).

12. [5, . 116] u y

u y end posy(u).

1. [5, 2.3.1] u, v Fact(y) |u| |v|. :

u v, Ry(v) Ry(u);

Ry(v) = Ry(u), end posy(u) = end posy(v) u Suff(v).

2. [5, 2.3.2] u, v, w Fact(y). u

v, v w, u y w, u y v y w.

3. [5, 2.3.3] u, v .

u v ,

:

Ry(u) Ry(v);

Ry(v) Ry(u);

Ry(u) Ry(v) = .

-

, (-

).

13.

C y Repry(C) = {x Fact(y) | Ry(x) = C}.

-

.

6 14.

C y Repry(C), -

.

15.

C y Repry(C), -

.

4.

,

. :

wmax wmin -

C y. , s Repry(C)

, s wmax |s| |wmin|.

s Repry(C). , s Suff(wmax) |s| |wmin|.

14 wmax Repry(C) |wmax| |s|. 1

, s Suff(wmax). , 15

|s| |wmin|, .

s Suff(wmax) |s| |wmin|. , s Repry(C).

15 1 wmin Suff(s). 14

15 , wmax y wmin. , 2 wmax y s y wmin,

.

.

5. -

.

Ry(x). wmax

, lenmin . ,

4 Repry(Ry(x)) = {z | z Suff(wmax) |z| lenmin}.

6. C len -

y len, , Rw(y) = C.

, y1 y2 len ,

C = Rw(y1) = Rw(y2). , y1 = y2. 1 y1

Suff(y2), y2 Suff(y1). , y1 y2

, , , y1 = y2,

.

71.3.

16. [5, . 118] Fact(w)

sw, w,

x Fact(w) \ {} x

x.

7. x2 = sw(x1). -

Rw(x2) Rw(x1).

16 , x2 x1

|x2| < |x1|.

y, x1, -

x2. y = x1[|x1| |x2|

1 . . . |x1|].

16 , y w x1 x2

Rw(x2). 2 , y 6w x1, ,

y Rw(y) = Rw(x1).

|y| = 1 + |x2|, .

8. x Fact(w) -

Rw(x) Rw(sw(x)).

16 sw(x) x, |sw(x)| < |x|

Rw(x) 6= Rw(sw(x)).

-

1.

1.4.

17. T w -

. -

( ). ,

, . |w|+1

, , . -

, (-

)

() w.

8 18. [1, . 122]

,

.

.

19. [1, . 122] v -

v v.

20. v -

.

21. u

RT (u) , -

u , -

u.

22. , v

Rw(x), RT (v) = Rw(x).

23. v u ,

RT (v) = RT (u).

. 1 aabb. -

.

b b

b b

b b

a

a

. 1. aabb

9. v -

v.

9 v x. , RT (v) =

Rw(x).

, y RT (v) , y Rw(x). y

RT (v) , xy w, 10

, y Rw(x).

10. w -

, , .

Rw(x) . Rw(x) ,

x w. , x w, ,

v x.

9.

1.5.

() ([5,

. 108], [1, . 121]). -

.

24. [5, . 108]

w, , ,

, .

. 2 aabb.

.

25. [1, . 122] ,

(u, v), (u, v) .

u (u, v)

, .

. 10 ,

w

( ).

10

abb

bb

b b

a

. 2. aabb

1.6.

26. [5, . 121] A w

-

, w .

. 3 aabb.

b

b

b

b b

a

a

. 3. aabb

27. , s -

Rw(x), RA(s) = Rw(x).

11. [5]

-

w.

11

, , -

, ,

.

12. , -

s, -

.

, -

, -

26, 13 9.

28. s , -

x.

s , s -

, s(x).

suffix(s).

29. s

reprmax(s).

. , -

([5, . 126]) ( -

- -

), , ,

, -

.

13. s -

. s

1 + reprmax(suffix(s)).

28 29,

7.

14. x -

s1 . s2 ,

, x[2 . . . |x|].

:

|x| = reprmax(suffix(s1)) + 1, s2 suffix(s1)

12

|x| > reprmax(suffix(s1)) + 1, s2 s1

, |x| > reprmax(suffix(s1))+1. 28

16 , x[2 . . . |x|] Rw(x). -

, , x[2 . . . |x|],

s1.

, |x| = reprmax(suffix(s1))+1. ,

x[2 . . . |x|] Rw(sw(x)), 16 -

, , x[2 . . . |x|],

suffix(s1).

30. [5, . 132] , s -

, -

:

s

s

1.7.

31. [5, . 132] A

w +,

, -

.

. 4 aabb.

b b

a bb

abb

. 4. aabb

13

15. A w.

A x s1 s2.

x = w[|w| |y| |x|+ 1 . . . |w| |y|], y RA(s2).

, A x

s1 s2 y RA(s2), , xy RA(s1). ,

, , xy Suff(w).

xy w,

xy = w[|w| |xy|+ 1 . . . |w|].

, x = w[|w| |xy| + 1 . . . |w| |y|] = w[|w| |x| +

|y|+ 1 . . . |w| |y|], .

32. longest(s) = maxxRA(s) |x|, -

, , -

s.

33. -

fork, :

s , fork(s) = s

s , fork(s) = t, t -

, s

16. w.

c s1 s2. :

w -

s1 fork(s2) y,

c.

y = w[|w| longest(s2) . . . |w| longest(fork(s2))].

, s2

fork(s2), , , , -

.

-

31.

, 1 +

longest(s2) = |y|+ longest(fork(s2)), 15.

14

, w -

fork longest,

O(|w|).

15

2.

2.1.

34. , u

s , RT (u) = RA(s).

. ,

-

, -

.

17. w -

w.

-

26.

s

, .

.

.

,

.

6, -

. , ,

16

, s,

.

, s

( ).

.

18. -

s, |r|, s

, r s.

2.2.

19. [5, . 132] -

.

20. [5, . 132]

w

w.

21. -

,

.

19.

. ,

-

.

, -

, ,

s, |r|, s

, r s.

17

2.3.

.

-

.

1 public SuffixTrie buildTrie(SuffixAutomaton auto)

{

2 SuffixTrie = new SuffixTrie();

3 int root = walk(trie , auto , auto.

getStartState ());

4 trie.setRootNode(root);

5 return trie;

6 }

7

8 private int walk(SuffixTrie trie ,

9 SuffixAutomaton auto ,

10 int state)

11 {

12 int node = trie.createNode();

13 if (auto.isFinal(state)) {

14 trie.setLeaf(node , true);

15 }

16 for ( Edge edge: auto.getEdges(state)) {

17 int newState = fork(edge.getTarget());

18 int newNode = walk(trie , auto , newState);

19 trie.addEdge(node , newNode , edge.getChar

());

18

20 }

21 return node;

22 }

, buildTrie(A)

A. , walk(T , A, s)

T , s A,

.

-

x, x RA(s) , x

walk .

.

x = . RA(s) , s -

. 13-15 , node

, RA(s).

.

, x 6= . RA(s) -

, A,

s.

, x RA(s), -

T x.

x RA(s) , A, s, -

x. , , , A

s x[1]. ,

16-20. , -

newNode x[2 . . . |x|]. ,

x.

, x 6 RA(s), -

T x.

. , -

x. , node

x[1]. 19. -

, newNode

x[2 . . . |x|], ,

A x[2 . . . |x|], s2, -

, x, s,

19

x 6 RA(s). , ,

node x .

. , -

O(n), n .

2.4.

.

.

1 public SuffixTree buildTree(

CompactSuffixAutomaton auto) {

2 SuffixTree = new SuffixTree();

3 int root = walk(tree , auto , auto.

getStartState ());

4 tree.setRootNode(root);

5 return tree;

6 }

7

8 private int walk(SuffixTree tree ,

9 CompactSuffixAutomaton auto ,

10 int state)

11 {

12 int node = tree.createNode();


14 tree.setLeaf(node , true);

15 }

20


17 int newNode = walk(tree , auto , edge.

getTarget());

18 tree.addEdge(node , newNode ,

19 edge.getBegin(),

20 edge.getEnd ());

21 }

22 return node;

23 }

, ,

, , . ,

. ,

, , -

, .

.

22. ,

walk, , .

, node, walk,

, , A

state .

, . ,

, , 13-15 node

, .

. O(n),

n .

, ,

.

, , -

, ,

,

w.

21

15, -

w -

.

.

fork longest.

2.

-

.

1 public SuffixTree buildTree(SuffixAutomaton auto)

{

2 SuffixTree = new SuffixTree();

3 int root = walk(tree , auto , auto.

getStartState ());

4 tree.setRootNode(root);

5 return tree;

6 }

7

8 private int walk(SuffixTree tree ,

9 SuffixAutomaton auto ,

10 int state)

11 {

12 int node = tree.createNode();


14 tree.setLeaf(node , true);

15 }


17 int newState = fork(edge.getTarget());

22

18 int b = length () - longest(edge.getTarget

()) - 1;

19 int e = length () - longest(newState);

20 int newNode = walk(tree , auto , newState);

21 tree.addEdge(node , newNode , b, e);

22 }

23 return node;

24 }

, .

2.5.

,

w :

w O(|w|);

, s

fork(s) longest(s);

, -

.

, w

:

;

,

.

, -

, .

23

, , -

O(|w|) fork

longest, , , .

. 1 , -

.

,

.

1.

{a, b}

. .

100000 0.177 0.125

200000 0.388 0.266

300000 0.596 0.428

400000 0.819 0.580

500000 1.035 0.730

600000 1.264 0.884

700000 1.470 1.047

800000 1.694 1.216

900000 1.923 1.342

1000000 2.153 1.502

.

-

, -

24

. ,

,

.

, ,

,

.

25

3.

-

. , -

. , -

.

3.1.

, :

;

, ;

;

;

.

-

.

,

-

, .

, -

, , .

26

, O(|w|) -

. ,

, .

-

. -

.

, -

. -

. v1 v2

, :

v1 v2 s1 s2 ,

s1 s2;

v1 v2 ,

v1 v2.

, .

, -

num , ,

lenmin ,

len

num + len lenmin.

, O(|w|) -

O(1).

.

3.1.1.

, v -

s , RT (v) =

Suff(w) = RA(s).

, , -

, 0 , -

.

27

3.1.2.

, -

.

v

, RT (v).

s ,

v. v ,

RA(s). , , , s -

.

, v -

, s

.

-

,

O(1).

3.1.3.

.

v .

14 , :

len v

s, ,

s, len 1;

-

s, , -

suffix(s), len 1.

, O(1).

3.1.4.

v

, c.

28

x v.

v s -

.

s

c, xc w, , ,

, v ,

c.

,

s t c. f = fork(t). -

31 33 ,

s f .

, 21 16 , -

s f -

y, c. -

, y = w[|w|longest(s2) . . . |w|longest(fork(s2))].

, fork longest,

v , -

c, , O(1)

c,

.

3.2.

(LCP ) .

1 s , -

.

si =

s[i . . . |s|] sj = s[j . . . |s|].

,

O(|s|) , -

O(log |s|) O(1).

29

,

:

;

;

k ;

;

.

LCP

.

s. , -

.

-

.

s,

.

, ,

-

.

, ,

,

3.1.

[6],

O(log |s|).

:

s;

30

-

.

, -

;

.

:

t1 t2

;

, -

, t1 t2;

.

, ,

. , :

s

;

.

-

2.

:

1 public void init(String str) {

2 globalTime = 0;

3 time2depth = new int[4 * str.length () + 1];

4 suffix2time = new int[str.length () + 1];

5 SuffixAutomaton auto = new SuffixAutomaton (

str);

6 walk(auto , auto.getStartState () , 0);

31

7 }

8

9 private void walk(SuffixAutomaton auto ,

10 int state ,

11 int depth)

12 {


14 suffix2time[depth ] = globalTime;

15 }

16 time2depth[globalTime ++] = depth;


18 int target = edge.getTarget();

19 int nextState = fork(target);

20 dfs(nextState , depth + 1 +

21 longest(target) - longest(nextState))

;

22 time2depth[globalTime ++] = depth;

23 }

24 }

, -

time2depth suffix2time. , walk

.

, -

. ,

walk node -

.

node , -

state . , node -

, , depth.

14 suffix2time , -

depth.

32

16 time2depth -

node.

17 - 23 node

. node, -

time2depth 22.

. 2 ,

, 1.3 - 1.9 .

2. -

LCP .

O(log |w|).

{a, b}

. .

100000 0.163 0.234

200000 0.323 0.495

300000 0.511 0.769

400000 0.683 1.017

500000 0.852 1.289

600000 1.039 1.552

700000 1.213 1.826

800000 1.381 2.103

900000 1.563 2.381

1000000 1.735 2.631

-

,

.

33

, -

,

. -

.

34

.

-

.

-

( -) -

, . , -

. -

,

, .

, -

. -

,

.

,

,

-

.

-

. , -

.

, -

.

, :

, ,

. , -

, , -

.

35

1. . , . -

. .: ; -

, 2003.

2. Weiner P. Linear pattern matching algorithms / Proc. of the 14th IEEE

Symp. on Swithing and Automata Theory. 1973, pp.1-11.

3. McCreight E. M. A space-economical sux tree construction alorithm // J.

ACM. 1976. Vol 23, pp. 262-272.

4. Ukkonen E. Online construction of sux-trees // Algoritmica. 1995. Vol. 14,

pp. 249-260.

5. Lothaire M. Applied Combinatorics on Words // Encyclopedia of

Mathematics and its Applications, 2005. Vol. 90. Cambridge University

Press, Cambridge.

6. Bender M., Farach-Colton M. The LCA Problem Revisited / LATIN 2000,

pp. 88-94.

7. ., ., . -

, . .: , 2002.

8. ., ., . : .

.: , 2000.

9. Sartaj Sahni Dr. Data Structures, Algorithms, & Applications in

Java. Sux Trees. CISE Department Chair at University of Florida.

http://www.cise.u.edu/sahni/dsaaj/enrich/c16/sux.htm

10. Edelkamp S. Sux tree // Dictionary of Algorithms and Data

Structures. U.S. National Institute of Standards and Technology. 2007.

http://www.nist.gov/dads/HTML/suxtree.html

36

11. Blumer A., Blumer J., Ehrenfeucht A., Hausler D., McConnel R. Linear

size nite automata for the set of all subwords of a word: an outline of results

// Bull. Eur. Assoc. Theoret. Commput. Sci. 1983. Vol 21, pp. 12-20.

12. Blumer A., Blumer J., Ehrenfeucht A., Hausler D., McConnel R. The

smallest automaton recognizing the subwords of a text // Bull. Eur. Assoc.

Theoret. Commput. Sci. 1985. Vol 40(1), pp. 31-55.

13. Eilenberg S. Automata, Languages, and Machines. Vol A. Academic Press.

1974.

14. Kuich W., Salomaa A. Semirings, Automata, Languages. Springer-Verlag.

1986.

15. Alonso L., Remy J. L., Schott R. A linear-time algorithm for the generation

of trees // Algorithmica. 1997. Vol 17(2), pp. 162-182.

16. Devroye L., Szpankowski W., Rais B. A note of the height of sux trees

// SIAM J. Comput. 1992. Vol. 21, pp. 48-53.

17. Farach M. Optimal sux tree construction with large alphabets // In 38th

Foundations of Computer Science (FOCS). 1997, pp. 137-143.

Documents

Suff Automata